Arturo Vidal scored the winner - and his second of the game - in the 101st minute after Luis Gutierrez was adjudged to have handled the ball when it struck him on the upper arm, with his hand tucked behind his back and his face turned away in the Group D encounter on Friday.
The decision sparked outrage from Bolivia, but Pizzi insisted that regardless of the means, the defending champions were worthy of the win and a crucial three points in Foxborough, Massachusetts.
"I stress the validity of this win," said Pizzi.
"It was a widely deserved victory. We had chances, lots of control and everything you need to have to win. We lacked precision and more intensity in the last section, but the second half was a little closer to what we want."
Earlier, Bolivia had equalised through substitute Jhasmani Campos, who curled a wonderful free-kick beyond Claudio Bravo after Vidal opened the scoring just a minute after the half-time break.
Vidal's late winner may have broken Bolivian hearts, but Pizzi had already moved onto stressing the importance of converting the win into knockout football by beating Panama on Tuesday.
Pizzi said he expects, "major difficulties [against Panama]. And we will overcome.
"We need to improve some things, but we have to keep believing we can achieve this improvement and we can impose our game."